6,818 research outputs found

    Profinite Galois descent in K(h)-local homotopy theory

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    We investigate the category of K(h)-local spectra through the action of the Morava stabiliser group. Using condensed mathematics, we give a model for the continuous action of this profinite group on the ∞-category of K(h)-local modules over Morava E-theory, and explain how this gives rise to descent spectral sequences computing the Picard and Brauer groups of K(h)-local spectra. In the second part, we focus on the computation of these spectral sequences at height one, showing that they recover the Hopkins-Mahowald-Sadofsky computation of the Picard group, and giving a complete computation of the Brauer group relative to p-completed complex K-theory

    On the stability of Fractal interpolation functions with variable parameters

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    Fractal interpolation function (FIF) is a fixed point of the Read–Bajraktarević operator defined on a suitable function space and is constructed via an iterated function system (IFS). In this paper, we considered the generalized affine FIF generated through the IFS defined by the functions Wn(x,y)=(an(x)+en,αn(x)y+ψn(x)) W_n(x, y) = \big(a_n(x)+e_n, \alpha_n(x) y +\psi_n(x)\big) , n=1,,N n = 1, \ldots, N . We studied the shift of the fractal interpolation curve, by computing the error estimate in response to a small perturbation on αn(x) \alpha_n(x) . In addition, we gave a sufficient condition on the perturbed IFS so that it satisfies the continuity condition. As an application, we computed an upper bound of the maximum range of the perturbed FIF

    Colossal Trajectory Mining: A unifying approach to mine behavioral mobility patterns

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    Spatio-temporal mobility patterns are at the core of strategic applications such as urban planning and monitoring. Depending on the strength of spatio-temporal constraints, different mobility patterns can be defined. While existing approaches work well in the extraction of groups of objects sharing fine-grained paths, the huge volume of large-scale data asks for coarse-grained solutions. In this paper, we introduce Colossal Trajectory Mining (CTM) to efficiently extract heterogeneous mobility patterns out of a multidimensional space that, along with space and time dimensions, can consider additional trajectory features (e.g., means of transport or activity) to characterize behavioral mobility patterns. The algorithm is natively designed in a distributed fashion, and the experimental evaluation shows its scalability with respect to the involved features and the cardinality of the trajectory dataset

    Deep Learning Techniques for Electroencephalography Analysis

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    In this thesis we design deep learning techniques for training deep neural networks on electroencephalography (EEG) data and in particular on two problems, namely EEG-based motor imagery decoding and EEG-based affect recognition, addressing challenges associated with them. Regarding the problem of motor imagery (MI) decoding, we first consider the various kinds of domain shifts in the EEG signals, caused by inter-individual differences (e.g. brain anatomy, personality and cognitive profile). These domain shifts render multi-subject training a challenging task and impede robust cross-subject generalization. We build a two-stage model ensemble architecture and propose two objectives to train it, combining the strengths of curriculum learning and collaborative training. Our subject-independent experiments on the large datasets of Physionet and OpenBMI, verify the effectiveness of our approach. Next, we explore the utilization of the spatial covariance of EEG signals through alignment techniques, with the goal of learning domain-invariant representations. We introduce a Riemannian framework that concurrently performs covariance-based signal alignment and data augmentation, while training a convolutional neural network (CNN) on EEG time-series. Experiments on the BCI IV-2a dataset show that our method performs superiorly over traditional alignment, by inducing regularization to the weights of the CNN. We also study the problem of EEG-based affect recognition, inspired by works suggesting that emotions can be expressed in relative terms, i.e. through ordinal comparisons between different affective state levels. We propose treating data samples in a pairwise manner to infer the ordinal relation between their corresponding affective state labels, as an auxiliary training objective. We incorporate our objective in a deep network architecture which we jointly train on the tasks of sample-wise classification and pairwise ordinal ranking. We evaluate our method on the affective datasets of DEAP and SEED and obtain performance improvements over deep networks trained without the additional ranking objective

    Quantum-Classical hybrid systems and their quasifree transformations

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    The focus of this work is the description of a framework for quantum-classical hybrid systems. The main emphasis lies on continuous variable systems described by canonical commutation relations and, more precisely, the quasifree case. Here, we are going to solve two main tasks: The first is to rigorously define spaces of states and observables, which are naturally connected within the general structure. Secondly, we want to describe quasifree channels for which both the Schrödinger picture and the Heisenberg picture are well defined. We start with a general introduction to operator algebras and algebraic quantum theory. Thereby, we highlight some of the mathematical details that are often taken for granted while working with purely quantum systems. Consequently, we discuss several possibilities and their advantages respectively disadvantages in describing classical systems analogously to the quantum formalism. The key takeaway is that there is no candidate for a classical state space or observable algebra that can be put easily alongside a quantum system to form a hybrid and simultaneously fulfills all of our requirements for such a partially quantum and partially classical system. Although these straightforward hybrid systems are not sufficient enough to represent a general approach, we use one of the candidates to prove an intermediate result, which showcases the advantages of a consequent hybrid ansatz: We provide a hybrid generalization of classical diffusion generators where the exchange of information between the classical and the quantum side is controlled by the induced noise on the quantum system. Then, we present solutions for our initial tasks. We start with a CCR-algebra where some variables may commute with all others and hence generate a classical subsystem. After clarifying the necessary representations, our hybrid states are given by continuous characteristic functions, and the according state space is equal to the state space of a non-unital C*-algebra. While this C*-algebra is not a suitable candidate for an observable algebra itself, we describe several possible subsets in its bidual which can serve this purpose. They can be more easily characterized and will also allow for a straightforward definition of a proper Heisenberg picture. The subsets are given by operator-valued functions on the classical phase space with varying degrees of regularity, such as universal measurability or strong*-continuity. We describe quasifree channels and their properties, including a state-channel correspondence, a factorization theorem, and some basic physical operations. All this works solely on the assumption of a quasifree system, but we also show that the more famous subclass of Gaussian systems fits well within this formulation and behaves as expected

    Attractors for a fluid-structure interaction problem in a time-dependent phase space

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    This paper is concerned with the long-time dynamics of a fluid-structure interaction problem describing a Poiseuille inflow through a 2D channel containing a rectangular obstacle. Physically, this models the interaction between the wind and the deck of a bridge in a wind tunnel experiment, as time goes to infinity. Due to this interaction, the fluid domain depends on time in an unknown fashion and the problem needs a delicate functional analytic setting. As a result, the solution operator associated to the system acts on a time- dependent phase space, and it cannot be described in terms of a semigroup nor of a process. Nonetheless, we are able to extend the notion of global attractor to this particular setting, and prove its existence and regularity. This provides a strong characterization of the asymptotic behavior of the problem. Moreover, when the inflow is sufficiently small, the attractor reduces to the unique stationary solution of the system, corresponding to a perfectly symmetric configuration

    On the Global Topology of Moduli Spaces of Riemannian Metrics with Holonomy Sp(n)\operatorname{Sp}(n)

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    We discuss aspects of the global topology of moduli spaces of hyperkähler metrics. If the second Betti number is larger than 44, we show that each connected component of these moduli spaces is not contractible. Moreover, in certain cases, we show that the components are simply connected and determine the second rational homotopy group. By that, we prove that the rank of the second homotopy group is bounded from below by the number of orbits of MBM-classes in the integral cohomology. \\ An explicit description of the moduli space of these hyperkähler metrics in terms of Torelli theorems will be given. We also provide such a description for the moduli space of Einstein metrics on the Enriques manifold. For the Enriques manifold, we also give an example of a desingularization process similar to the Kummer construction of Ricci-flat metrics on a Kummer K3K3 surface.\\ We will use these theorems to provide topological statements for moduli spaces of Ricci-flat and Einstein metrics in any dimension larger than 33. For a compact simply connected manifold NN we show that the moduli space of Ricci flat metrics on N×TkN\times T^k splits homeomorphically into a product of the moduli space of Ricci flat metrics on NN and the moduli of sectional curvature flat metrics on the torus TkT^k

    Mean curvature flow with generic initial data II

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    We show that the mean curvature flow of a generic closed surface in R3\mathbb{R}^3 avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical end cannot arise generically.Comment: Comments welcome

    Medical Image Analysis using Deep Relational Learning

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    In the past ten years, with the help of deep learning, especially the rapid development of deep neural networks, medical image analysis has made remarkable progress. However, how to effectively use the relational information between various tissues or organs in medical images is still a very challenging problem, and it has not been fully studied. In this thesis, we propose two novel solutions to this problem based on deep relational learning. First, we propose a context-aware fully convolutional network that effectively models implicit relation information between features to perform medical image segmentation. The network achieves the state-of-the-art segmentation results on the Multi Modal Brain Tumor Segmentation 2017 (BraTS2017) and Multi Modal Brain Tumor Segmentation 2018 (BraTS2018) data sets. Subsequently, we propose a new hierarchical homography estimation network to achieve accurate medical image mosaicing by learning the explicit spatial relationship between adjacent frames. We use the UCL Fetoscopy Placenta dataset to conduct experiments and our hierarchical homography estimation network outperforms the other state-of-the-art mosaicing methods while generating robust and meaningful mosaicing result on unseen frames.Comment: arXiv admin note: substantial text overlap with arXiv:2007.0778

    Quantum ergodicity on the Bruhat-Tits building for PGL(3,F)\text{PGL}(3, F) in the Benjamini-Schramm limit

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    We study eigenfunctions of the spherical Hecke algebra acting on L2(Γn\G/K)L^2(\Gamma_n \backslash G / K) where G=PGL(3,F)G = \text{PGL}(3, F) with FF a non-archimedean local field of characteristic zero, K=PGL(3,O)K = \text{PGL}(3, \mathcal{O}) with O\mathcal{O} the ring of integers of FF, and (Γn)(\Gamma_n) is a sequence of cocompact torsionfree lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself.Comment: 111 pages, 25 figures, 2 table
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